The hidden truth of ideal relativistic fluids: the matter Lagrangian is its total energy density, not its pressure!

TL;DR

This paper clarifies that for ideal relativistic fluids, the matter Lagrangian must be the total energy density rather than pressure, resolving previous inconsistencies in coupling with gravity.

Contribution

It demonstrates that the matter Lagrangian of an ideal fluid is physically consistent only when identified with its total energy density, not pressure.

Findings

Identifying Lagrangian with pressure causes inconsistencies.

Coupling to gravity requires Lagrangian to be total energy density.

Clarifies the correct form of matter Lagrangian for ideal fluids.

Abstract

It has been established in the literature that the matter Lagrangian of an ideal fluid can be expressed either as its total energy density or as its pressure. In this work, we demonstrate that identifying the matter Lagrangian with the pressure leads to physical inconsistencies, which are resolved when the fluid is coupled to the gravitational field. In such a scenario, the matter Lagrangian necessarily assumes the value of the total energy density. We thus conclude that, for an ideal fluid, the only physically consistent choice for the matter Lagrangian is its total energy density.